¿x^3/2/x-6=x-8?

x^(3/2) / (x-6) =x-8;  elevo ambos lados al cuadrado, manteniéndose la igualdad:

x^3 / (x^2-12x+36) = x^2-16x+64;

x^3 = (x^2-12x+36) * (x^2-16x+64);

x^3 = x^4 - 28x^3 +292x^2 -1344x + 2304;

0 = x^4 - 29x^3 +292x^2 -1344x + 2304;

Puede resolverse mediante Ferrari, pero hoy en día no se suele usar y se obtiene mediante iteraciones como Newton-Rapson o, como lo hice, mediante programas matemáticos como Wolfram, Derive u otros.

Una solución real es igual a 4, y tienes además otra real y dos complejas conjugadas que puedes ver en:

https://www.wolframalpha.com/input/?i=0+%3D+x%5E4+...

Corroboremos a x=4 con la igualdad inicial:

4^(3/2) / (4-6) = 4-8;

8 / (-2) = (-4);

(-4) = (-4);  es correcto.

x^(3/2) / (x-6) = x-8 ⇔ x^(3/2) = (x-6)(x-8)   

Te conviene realizar un ábaco. 

Representar las dos funciones y ver aproximadamente los puntos de corte.

f(x) = x^(3/2)

g(x) = (x-6)(x-8)

Se ve que tiene solamente dos raíces reales la ecuación: x^(3/2) = (x-6)(x-8)  

Una cerca de 4 y otra cerca del 14

Para encontrar aproximación mas exacta usas el metodo de "Newton" y "Partes proporcionales", una vez cada uno.

O lo sacas de la interseccion de las dos graficas.

x=4

o 

x≈14.496160894012 ...

Ver las soluciones en:

https://www.wolframalpha.com/input/?i=0+%3D+x%5E%2...

PD 

Si elevas al cuadrado, estas introduciendo raíces extrañas.

En este caso salen de la ecuación:

 x^(3/2) = - (x-6)(x-8)   

Que tiene dos raíces imaginarias  

La opuesta a la gráfica de f(x) no se corta corta con la función g(x)  

x^(3/2) / (x - 6) = x - 8

x^(3/2) = (x - 6).(x - 8)

x^(3/2) = x² - 8x - 6x + 48

x^(3/2) = x² - 14x + 48

[x^(3)]^(1/2) = x² - 14x + 48

x³ = (x² - 14x + 48)²

x³ = x⁴ - 28x³ + 292x² - 1344x + 2304

x⁴ - 29x³ + 292x² - 1344x + 2304 = 0

x⁴ - (25x³ + 4x³) + (192x² + 100x²) - (576x + 768x) + 2304 = 0

x⁴ - 25x³ - 4x³ + 192x² + 100x² - 576x - 768x + 2304 = 0

x⁴ - 25x³ + 192x² - 576x - 4x³ + 100x² - 768x + 2304 = 0

[x⁴ - 25x³ + 192x² - 576x] - [4x³ - 100x² + 768x - 2304] = 0

x.[x³ - 25x² + 192x - 576] - 4.[x³ - 25x² + 192x - 576] = 0

(x - 4).(x³ - 25x² + 192x - 576) = 0

First case: (x - 4) = 0 → x = 4

Second case: (x³ - 25x² + 192x - 576) = 0

x³ - 25x² + 192x - 576 = 0 ← it's necessary to eliminate the term at the power 2

x³ - 25x² + 192x - 576 = 0 → let: x = z + (25/3)

[z + (25/3)]³ - 25.[z + (25/3)]² + 192.[z + (25/3)] - 576 = 0

[z + (25/3)]².[z + (25/3)] - 25.[z² + (50/3).z + (25/3)²] + 192z + 1600 - 576 = 0

[z² + (50/3).z + (25/3)²].[z + (25/3)] - 25z² - (1250/3).z - (25³/9) + 192z + 1024 = 0

[z³ + (25/3).z² + (50/3).z² + (1250/9).z + (625/9).z + (25/3)³] - 25z² - (674/3).z - (6409/9) = 0

[z³ + 25z² + (1875/9).z + (15625/27)] - 25z² - (674/3).z - (6409/9) = 0

z³ + 25z² + (1875/9).z + (15625/27) - 25z² - (674/3).z - (6409/9) = 0

z³ - (147/9).z - (3602/27) = 0

z³ - (49/3).z - (3602/27) = 0 ← no term with power 2

z³ - (49/3).z - (3602/27) = 0 → let: z = u + v

(u + v)³ - (49/3).(u + v) - (3602/27) = 0

[(u + v)².(u + v)] - (49/3).(u + v) - (3602/27) = 0

[(u² + 2uv + v²).(u + v)] - (49/3).(u + v) - (3602/27) = 0

[u³ + u²v + 2u²v + 2uv² + uv² + v³] - (49/3).(u + v) - (3602/27) = 0

[u³ + v³ + 3u²v + 3uv²] - (49/3).(u + v) - (3602/27) = 0

[(u³ + v³) + (3u²v + 3uv²)] - (49/3).(u + v) - (3602/27) = 0

[(u³ + v³) + 3uv.(u + v)] - (49/3).(u + v) - (3602/27) = 0

(u³ + v³) + 3uv.(u + v) - (49/3).(u + v) - (3602/27) = 0 → you can factorize: (u + v)

(u³ + v³) + (u + v).[3uv - (49/3)] - (3602/27) = 0 → suppose that: [3uv - (49/3)] = 0 ← equation (1)

(u³ + v³) + (u + v).[0] - (3602/27) = 0

(u³ + v³) - (3602/27) = 0 ← equation (2)

You can get a system of 2 equations:

(1) : [3uv - (49/3)] = 0

(1) : uv = 49/9

(1) : u³v³ = (49/9)³

(2) : (u³ + v³) - (3602/27) = 0

(2) : u³ + v³ = 3602/27

Let: U = u³

Let: V = v³

You can get a new system of 2 equations:

(1) : UV = (49/9)³ ← this is the product P

(2) : U + V = 3602/27 ← this is the sum S

You know that the values U & V are the solutions of the following equation:

x² - Sx + P = 0 ← don’t confuse with the item x (initial equation)

x² - (3602/27).x + (49/9)³ = 0

Δ = (3602/27)² - [4 * (49/9)³]

Δ = (3602²/27²) - (4 * 49³/9³) → you know that: 9³ = 27²

Δ = [3602² - (4 * 49³)]/27²

Δ = 12503808/27²

Δ = (67 * 432²)/27²

Δ = 67 * (432/27)²

U = [(3602/27) + (432/27)√67]/2 = (1801 + 216√67)/27 → recall: U = u³ → u = U^(1/3)

u = [(1801 + 216√67)/27]^(1/3)

V = [(3602/27) - (432/27)√67]/2 = (1801 - 216√67)/27 ← this is V → recall: V = v³ → v = V^(1/3)

v = [(1801 - 216√67)/27]^(1/3)

Recall: z = u + v

Recall: x = z + (25/3)

x = u + v - (1/2)

x = [(1801 + 216√67)/27]^(1/3) + [(1801 - 216√67)/27]^(1/3) + (25/3)

x ≈ 5.09404053193468 + 1.06878702874723 + (25/3)

x ≈ 14.4961608940152

→ Solution = { 4 ; 14.4961608940152 }

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